A remark on the Hardy-Littlewood convergence test for Fourier series

Fourier Series Acceleration and Hardy-littlewood Series

We discuss the effects of the δ2 and Lubkin acceleration methods on the partial sums of Fourier Series. We construct continuous, even Hölder continuous functions, for which these acceleration methods fail to give convergence. The constructed functions include some interesting trigonometric series whose properties were investigated by Hardy and Littlewood.

Hardy-littlewood Type Inequalities for Laguerre Series

Let {cj} be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on {cj} to obtain the pointwise convergence as well as Lr -convergence of Laguerre series ∑ cj a j . Then, we prove aHardy-Littlewood type inequality ∫∞ 0 |f(t)|r dt≤ C ∑∞ j=0 |cj| j̄1−r/2 for certain r ≤ 1, where f is the limit function of ∑ cj a j . Moreover, we show that if f(x) ∼ ∑cj a j is ...

On the Hardy - Littlewood majorant

A Remark on Littlewood-paley Theory for the Distorted Fourier Transform

We consider the classical theorems of Mikhlin and LittlewoodPaley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator −∆+ V . We show that for such operators which display a zero energy resonance the full range 1 < p < ∞ in the Mikhlin theore...

The Hardy-Littlewood Function An Exercise in Slowly Convergent Series

The function in question is H(x) = ∑∞ k=1 sin(x/k)/k. In deference to the general theme of this conference, a summation procedure is first described using orthogonal polynomials and polynomial/rational Gauss quadrature. Its effectiveness is limited to relatively small (positive) values of x. Direct summation with acceleration is shown to be more powerful for very large values of x. Such values ...